Knot concordance: Fifty years since Fox and Milnor, June 2008

50年前，在1957年夏季AMS会议的一份研究公告中，R.H.福克斯和J.米尔纳开始了他们有影响力的合作，标题是《4-空间中2-球体的奇点和纽结的等价性》。在这里，他们提出了一个开创性的想法，即奇点链接的和谐性类别阻碍了它的移除。纽结的一致性和理解奇点的激励目标仍然是拓扑学和代数几何的核心。Brandeis大学将于2008年6月2日至5日举行一次会议，将几何拓扑学的各种研究人员和学生聚集在一起，他们的工作与这一基本思想有关。这次会议将通过鼓励数学互动和合作来丰富新的研究方向。将邀请大量的年轻调查人员给他们提供机会，开阔他们的视野，让他们相互了解，并了解这些领域的更资深成员。将有大约20个受邀地址，并将留出充足的时间让与会者之间的互动。会议还将纪念杰罗姆·莱文，他是该领域的先驱和关键贡献者。结是三维空间中不相交的闭合曲线。闭合的意思是，如果一个人沿着曲线行进，从曲线上的任何一点出发，他最终会回到原点。作为拓扑学领域的基础学科，纽结理论也与几何学、生物学和物理学的重要领域相互作用。纽结协调组测量四维空间中曲面的褶皱或奇点，并通过这种联系启发我们对四维形状的理解。尽管在过去的50年里付出了巨大的努力，但这些群体还没有得到充分的计算。近年来，来自低维拓扑的新技术，包括规范理论、非对易代数、高阶链理论和量子拓扑学，使纽结协调的研究取得了很大进展，揭示了纽结协调群中的新结构。这次会议将促进致力于结协调这些方面的研究人员之间的互动，并产生对这一基本问题的未来研究。

Fifty years ago, in a 1957 research announcement for the summer AMS meeting, R. H. Fox and J. Milnor, began their influential collaboration with the title, `Singularities of 2-spheres in 4-space and equivalence of knots.' Here they introduced the seminal idea that the concordance class of the link of a singularity obstructs its removal. Both concordance of knots, and the motivating goal of understanding singularities remain central to topology and algebraic geometry. A conference at Brandeis University will be held on June 2-5, 2008, to bring together a variety of researchers and students in geometric topology whose work connects to this fundamental idea. The conference will fertilize new research directions by encouraging mathematical interaction and collaboration. A substantial number of young investigators will be invited to give them exposure, broaden their perspective, and allow them to get to know each other and the more senior members of these fields. There will be approximately twenty invited addresses, and ample time will be set aside for interaction among the attendees. The conference will also honor the memory of Jerome Levine, a pioneer and key contributor to the field.A knot is a non-intersecting closed curve in three dimensional space. By closed, one means that if one travels along the curve, originating from any point on the curve, one eventually returns to the point of origin. A foundational subject to the field of topology, knot theory interacts with important areas in geometry, biology, and physics as well. The Knot concordance group measures wrinkles, or singularities, of surfaces in four dimensional space, and through this connection, enlightens our understanding of four dimensional shapes. Despite intense effort over the last 50 years, these groups have not been fully computed. Recently, new techniques from low-dimensional topology, including gauge theory, non-commutative algebra, higher- order linking theory, and quantum topology, have resulted in great advances in the study of knot concordance, revealing new structures in the concordance group. The conference will promote interactions between researchers working on these aspects of knot concordance and engender future research on this fundamental problem.